This project grows out of the discovery of counterexamples to mathematical claims made by Ansley Coale in his 1972 book The Growth and Structure of Human Populations. Coale's Chapter 3 treats the "Convergence of a Population (Age Structure) to the Stable Form" under the deterministic theory of Lotka, in which transient waves in the age pyramid and the rates of their gradual disappearance are governed by the complex roots of Lotka's equation. Coale claims that the roots governing the wave of lowest frequency can always be moved to the imaginary axis of the complex plane by rescaling the level while preserving the shape of the net maternity function, and this rescaling property becomes the basis of his heuristic discussion of the distribution of the roots and their influences. Unfortunately, this claim is false in general, and important aspects of the standard account of approach to stability, which derives from Coale, therefore, come into doubt. This application proposes an analytical investigation of necessary conditions and of sufficient conditions for the rescaling property to hold. Not only the lowest frequency wave but all the waves will come under study. Computer generated examples of the behavior of roots under rescaling will guide the analysis. The aim is to repair the standard account of Lotka's roots and of population approach to stability, in order to put on a regorous basis what is already part of the fundamental textbook material of demography.